# Another question on Open and Closed in V .... D&K Proposition 1.2.17 .... ....

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In summary, Duistermaat and Kolk provide a proof for Proposition 1.2.17 regarding open and closed sets in a set V. However, there is a question about a rigorous proof for part (i). By Definition 1.2.16, if A is open in V, then there exists an open subset U of $\mathbb{R}^{n}$ such that A=V$\cap$U. Since both U and V are open in $\mathbb{R}^{n}$, A is also open in $\mathbb{R}^{n}$. Furthermore, if A is open in $\mathbb{R}^{n}$, then it is also open in V.
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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with another aspect of the proof of Proposition 1.2.17 ... ...

Duistermaat and Kolk's Proposition 1.2.17 and the preceding definition (regarding open and closed sets in a set V) read as follows:
View attachment 7735
View attachment 7736D&K write that the proof of (i) is immediate from the definitions ... but I have been unable to formulate a rigorous proof of (i) ... could someone please demonstrate a rigorous proof of (i) ... ...Hope that someone can help ...

Peter========================================================================================
D&K's definitions and early results on open and closed sets may be helpful to MHB members reading and following the above post ... so I am providing the same ... as follows:View attachment 7737
View attachment 7738Hope that helps ... ...

Peter

Last edited:
Hi, Peter.

Suppose $A$ is open in $V$. By Definition 1.2.16 there is an open subset, say U, of $\mathbb{R}^{n}$ such that $A=V\cap U.$ Since both $U$ and $V$ are open in $\mathbb{R}^{n}$ and finite intersections of open sets are open, $A$ is open in $\mathbb{R}^{n}.$

If $A$ is open in $\mathbb{R}^{n}$, then it is open in $V$ because $A=V\cap A$ (since $A\subseteq V$).

GJA said:
Hi, Peter.

Suppose $A$ is open in $V$. By Definition 1.2.16 there is an open subset, say U, of $\mathbb{R}^{n}$ such that $A=V\cap U.$ Since both $U$ and $V$ are open in $\mathbb{R}^{n}$ and finite intersections of open sets are open, $A$ is open in $\mathbb{R}^{n}.$

If $A$ is open in $\mathbb{R}^{n}$, then it is open in $V$ because $A=V\cap A$ (since $A\subseteq V$).
THanks GJA ... appreciate the help ...

Peter

## 1. What is the meaning of Open and Closed sets in V?

Open and Closed sets are important concepts in topology, a branch of mathematics that studies the properties of geometric spaces. In the context of V, Open and Closed sets refer to subsets of the vector space that have certain characteristics.

## 2. How are Open and Closed sets related to D&K Proposition 1.2.17?

D&K Proposition 1.2.17 is a specific theorem that discusses the relationship between Open and Closed sets in V. It states that in a topological vector space, a set is Closed if and only if its complement is Open.

## 3. Can you give an example of an Open set in V?

Yes, an example of an Open set in V is the set of all real numbers between 0 and 1, denoted as (0,1). This set includes all numbers between 0 and 1, but does not include the endpoints 0 and 1. Therefore, it is an Open set.

## 4. How can I identify if a set is Closed in V?

A set is Closed in V if it contains all its limit points. In other words, if every convergent sequence of elements in the set has a limit that is also contained in the set, then the set is Closed.

## 5. What is the significance of Open and Closed sets in V?

Open and Closed sets play a crucial role in analyzing the properties and behavior of topological vector spaces. They help in understanding convergence, continuity, and compactness of functions and transformations in V. They also have applications in other areas of mathematics, such as functional analysis and differential equations.

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